A360632 -id:A360632 - cp's OEIS frontend

A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

1, 2, 4, 3, 10, 21, 5, 22, 73, 192, 7, 44, 190, 703, 2035, 11, 91, 510, 2287, 8581, 27407, 15, 172, 1196, 6738, 30209, 118461, 399618, 22, 326, 2895, 19160, 102092, 462114

Offset: 1

Pontus von Brömssen, Feb 14 2023

Pieces are free to rotate by 90 degrees, i.e., an r X s piece and an s X r piece are equivalent. See A360451 for the case when the pieces are fixed.

			Triangle begins:
   n\k|  1   2    3    4     5      6      7
   ---+--------------------------------------
   1  |  1
   2  |  2   4
   3  |  3  10   21
   4  |  5  22   73  192
   5  |  7  44  190  703  2035
   6  | 11  91  510 2287  8581  27407
   7  | 15 172 1196 6738 30209 118461 399618
   ...
T(2,2) = 4, because a 2 X 2 rectangle can be tiled by: one 2 X 2 piece; two 1 X 2 pieces; one 1 X 2 piece and two 1 X 1 pieces; four 1 X 1 pieces.
The T(3,2) = 10 sets of pieces that can tile a 3 X 2 rectangle are shown in the table below. (Each column on the right gives a set of pieces.)
   length X width |  number of pieces
   ---------------+--------------------
        2 X 3     | 1 0 0 0 0 0 0 0 0 0
        2 X 2     | 0 1 1 0 0 0 0 0 0 0
        1 X 3     | 0 0 0 2 1 1 0 0 0 0
        1 X 2     | 0 1 0 0 1 0 3 2 1 0
        1 X 1     | 0 0 2 0 1 3 0 2 4 6

Cf. A000041 (column k=1), A116694, A224697 (square pieces), A360451 (fixed pieces), A360630 (main diagonal), A360631 (column k=2), A360632 (column k=3).

T(7,7) and T(8,k) for k = 1..6 added by Robin Visser, May 09 2025

A360631 Number of sets of integer-sided rectangular pieces that can tile a 2 X n rectangle.

Original entry on oeis.org

1, 2, 4, 10, 22, 44, 91, 172, 326, 595, 1066, 1849, 3204, 5365, 8921, 14581, 23558, 37440, 59127, 91957, 142060, 217015, 328939, 493917, 737249, 1090432, 1603439, 2341094, 3398313, 4900740, 7032809, 10031010, 14241165, 20112575, 28276657, 39566635, 55140425, 76499692, 105731884, 145550924

Offset: 0

Pontus von Brömssen, Feb 14 2023

nonn

			From _Robin Visser_, May 01 2025: (Start)
For n = 1, there are a(1) = 2 possible sets of rectangular pieces that can tile a 2 x 1 rectangle: one 1 x 2 piece; or two 1 x 1 pieces.
For n = 2, there are a(2) = 4 possible sets of rectangular pieces that can tile a 2 x 2 rectangle: one 2 x 2 piece; two 1 x 2 pieces; one 1 x 2 piece and two 1 x 1 pieces; or four 1 x 1 pieces.
For n = 3, there are a(3) = 10 possible sets of rectangular pieces that can tile a 2 x 3 rectangle: one 2 x 3 piece; one 2 x 2 piece and one 1 x 2 piece; one 2 x 2 piece and two 1 x 1 pieces; two 1 x 3 pieces; one 1 x 3 piece, one 1 x 2 piece, and one 1 x 1 piece; one 1 x 3 piece and three 1 x 1 pieces; three 1 x 2 pieces; two 1 x 2 pieces and two 1 x 1 pieces; one 1 x 2 piece and four 1 x 1 pieces; or six 1 x 1 pieces. (End)

Robin Visser, Table of n, a(n) for n = 0..40

Second column of A360629.

Cf. A000041 (1 x n rectangle), A360632 (3 x n rectangle).

def a(n):
    A, B = [set() for i in range(n+1)], [set() for i in range(n+1)]
    A[0].add(()); B[0].add(());
    for (m,k) in [(x,y) for x in range(1, n+1) for y in range(1, x+1)]:
        for p in A[m-k]: A[m].add(tuple(sorted(list(p)+[k])))
        for (p, q) in [(x,y) for x in A[m] for y in A[m]]:
            B[m].add(tuple(sorted([(1,c) for c in p]+[(1,c) for c in q])))
        for p in B[m-k]: B[m].add(tuple(sorted(list(p)+[tuple(sorted((2,k)))])))
    return len(B[n])  # Robin Visser, May 01 2025

More terms from Robin Visser, May 04 2025

a(0) = 1 prepended by Robin Visser, May 05 2025

A361219 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle.

Original entry on oeis.org

2, 11, 56, 370, 2666, 19126, 134902, 1026667, 8049132, 60996816, 450456500, 3427769018, 27127841200, 211563038980, 1837421211974, 15474223886906

Offset: 1

Pontus von Brömssen, Mar 05 2023

Tilings that are rotations or reflections of each other are considered distinct.

			The following table shows the sets of pieces that give the maximum number of tilings for n <= 16. All these sets are unique.
    \     Number of pieces of size
   n \  1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
  ----+-------+-------+-------+------
   1  |   1   |   1   |   0   |   0
   2  |   2   |   2   |   0   |   0
   3  |   3   |   3   |   0   |   0
   4  |   3   |   3   |   1   |   0
   5  |   4   |   4   |   1   |   0
   6  |   7   |   4   |   1   |   0
   7  |   8   |   5   |   1   |   0
   8  |   8   |   5   |   2   |   0
   9  |   9   |   6   |   2   |   0
  10  |  10   |   7   |   2   |   0
  11  |  11   |   8   |   2   |   0
  12  |  11   |   8   |   3   |   0
  13  |  12   |   9   |   3   |   0
  14  |  13   |   8   |   3   |   1
  15  |  14   |   9   |   3   |   1
  16  |  15   |  10   |   3   |   1

Third column of A361216.

Cf. A360632, A361225.

A361225 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle, up to rotations and reflections.

Original entry on oeis.org

1, 5, 8, 95, 682, 4801, 33807

Offset: 1

Pontus von Brömssen, Mar 05 2023

			The following table shows the sets of pieces that give the maximum number of tilings for n <= 7. The solutions are unique except for n = 1 and n = 3.
   \ Number of pieces of size
  n \  1 X 1 | 1 X 2 | 1 X 3
  ---+-------+-------+------
  1  |   3   |   0   |   0
  1  |   1   |   1   |   0
  1  |   0   |   0   |   1
  2  |   2   |   2   |   0
  3  |   3   |   3   |   0
  3  |   2   |   2   |   1
  4  |   3   |   3   |   1
  5  |   4   |   4   |   1
  6  |   7   |   4   |   1
  7  |   8   |   5   |   1
It seems that all optimal solutions for A361219 are also optimal here, but for n = 1 and n = 3 there are other optimal solutions.

Third column of A361221.

Cf. A361219, A360632.

A361000 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 3, 2, 4, 3, 5, 3, 6, 4, 6, 3, 9, 3, 6, 6, 8, 3, 9, 3, 10, 6, 6, 3, 13, 5, 6, 6, 10, 3, 13, 3, 10, 6, 6, 7, 15, 3, 6, 6, 14, 3, 13, 3, 10, 10, 6, 3, 17, 5, 10, 6, 10, 3, 13, 7, 14, 6, 6, 3, 21, 3, 6, 10, 12, 7, 13, 3, 10, 6, 14, 3, 21, 3, 6, 10, 10, 7, 13, 3

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Third column of A360998.

Cf. A000005, A083039, A360632, A361005.

A083039(n) = ([3, 1, 2, 2, 2, 1][n%6+1]);
A361000(n) = if(n<3, 1+n, 2*numdiv(n) - A083039(n)); \\ Antti Karttunen, Jul 19 2024

a(n) = 2*A000005(n) - A083039(n) for n >= 3.
Proof: (Start)
Note that a nonrearrangeable tiling must have full symmetry, otherwise it could be rotated or reflected to another tiling.
Identical pieces of size m X 1 tile the n X 3 rectangle in a unique way exactly when m is a divisor of n, except m = 2 and m = 3. Identical pieces of size m X 3 tile the rectangle in a unique way exactly when m is a divisor of n, except m = 1. The only other pieces that fit in the rectangle are of size m X 2, where m = 2 or m >= 4, but it is easily seen that the rectangle cannot be tiled with identical pieces of those sizes, so there are 2*A000005(n) - 1 - [n even] - [n divisible by 3] = 2*A000005(n) - A083039(n) nonrearrangeable tilings of the rectangle with identical pieces.
We now prove that there is no nonrearrangeble tiling of the rectangle that contains pieces of different sizes. Assume that we have such a tiling of the rectangle of height n and width 3, for the smallest possible value of n.
A fault of a tiling is a line through the interior of the rectangle that does not pass through the interior of any piece. If the tiling has any horizontal (or vertical) faults, all the resulting horizontal (or vertical) strips between the faults must have the same size and be tiled in the same way, otherwise the strips could be permuted to another tiling.
If the tiling has any vertical faults, there must be three identically tiled vertical n X 1 strips. This implies that a single strip must contain pieces of different lengths, which is clearly impossible because those pieces could be permuted to another tiling.
If there are no vertical faults, there must be a piece in the tiling with a horizontal side of length at least 2, and by symmetry one of its horizontal edges must be part of a horizontal fault, otherwise another tiling could be obtained by reflection. The resulting strips (of width 3) must all be tiled in the same way (and thus contain pieces of different sizes), and the tiling of one strip must be nonrearrangeble. We have thus obtained a nonrearrangeable tiling with pieces of different sizes for a smaller height, a contradiction.
(End)

A361005 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

Original entry on oeis.org

3, 7, 9, 18, 22, 42, 48, 86, 101

Offset: 1

Pontus von Brömssen, Feb 28 2023

Third column of A361001.

Cf. A360632, A361000.

A361427 Maximum difficulty level (see A361424 for the definition) for tiling an n X 3 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

Original entry on oeis.org

2, 6, 8, 48, 80, 480, 960, 1920, 3360, 13440, 20160, 60480, 80640, 201600, 967680, 1612800

Offset: 1

Pontus von Brömssen, Mar 11 2023

For all currently known terms, the maximum difficulty level is an integer.

			The following table shows all sets of pieces that give the maximum (n,3)-tiling difficulty level up to n = 16.
    \               Number of pieces of size
   n \ 1X1|1X2|1X3|1X4|1X5|1X6|1X7|1X8|1X9|1X10|1X12|2X2|2X3|2X4|2X5
  ----+---+---+---+---+---+---+---+---+---+----+----+---+---+---+---
   1  | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   1  | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   2  | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   3  | 1 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   4  | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   5  | 0 | 3 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   5  | 0 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 0 | 0
   6  | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0  | 0  | 1 | 0 | 0 | 0
   7  | 0 | 3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0  | 0  | 1 | 0 | 0 | 0
   7  | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0  | 0  | 0 | 1 | 0 | 0
   8  | 0 | 3 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0  | 0  | 0 | 1 | 0 | 0
   9  | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0  | 0  | 1 | 1 | 0 | 0
   9  | 2 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0  | 0  | 1 | 1 | 0 | 0
   9  | 0 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0  | 0  | 1 | 0 | 0 | 0
   9  | 0 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0  | 0  | 2 | 0 | 0 | 0
  10  | 3 | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 0  | 0  | 1 | 0 | 0 | 1
  11  | 0 | 2 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 1  | 0  | 0 | 1 | 0 | 0
  12  | 3 | 0 | 0 | 2 | 0 | 0 | 1 | 0 | 0 | 0  | 0  | 2 | 0 | 0 | 1
  13  | 1 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0 | 0  | 0  | 1 | 0 | 1 | 1
  13  | 0 | 2 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 0  | 1  | 1 | 1 | 0 | 0
  14  | 4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0  | 0  | 1 | 0 | 2 | 0
  14  | 3 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0  | 0  | 3 | 0 | 0 | 1
  14  | 2 | 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | 0  | 0  | 1 | 0 | 1 | 0
  15  | 1 | 0 | 0 | 3 | 2 | 1 | 0 | 0 | 0 | 0  | 0  | 0 | 1 | 0 | 1
  16  | 3 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | 0 | 0  | 0  | 0 | 0 | 2 | 1

Third column of A361424.

Cf. A360632, A361219, A361225.

A361525 Number of ways of dividing an n X 3 rectangle into integer-sided rectangles, up to rotations and reflections.

Original entry on oeis.org

1, 3, 17, 54, 892, 8159, 80021, 791821, 7906439, 79069308

Offset: 0

Pontus von Brömssen, Mar 15 2023

Third column of A361523.

Cf. A208215 (rotations and reflections are considered distinct), A359019 (square pieces), A360632.

a(n) >= A208215(n)/4 for n != 3.

a(n) ~ A208215(n)/4.

cp's OEIS Frontend

A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

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A360631 Number of sets of integer-sided rectangular pieces that can tile a 2 X n rectangle.

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A361219 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle.

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A361225 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle, up to rotations and reflections.

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A361000 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

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A361005 Number of tilings of an n X 3 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

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A361427 Maximum difficulty level (see A361424 for the definition) for tiling an n X 3 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

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A361525 Number of ways of dividing an n X 3 rectangle into integer-sided rectangles, up to rotations and reflections.

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